Democrats traditionally have an advantage with women voters. But Mr. Obama's advantage among women has grown to striking proportions. The president is also now even winning married women voters in the two states -- a group that previously backed Mitt Romney.

Romney, meanwhile, has an advantage among men in both swing states. But it's not wide enough to make up for his lack of female support. CBS News director of surveys Sarah Dutton said the data doesn't point to any one clear explanation for the widening of the gender gap.

"Women are much more likely to think Barack Obama cares about them than Romney does, and they think Barack Obama would do a better job than Mitt Romney on women's reproductive health issues, by two to one," Dutton said. "It's difficult to isolate just one reason for the increase in support among women."

In Ohio, women prefer Mr. Obama over Romney by a margin of 25 points, 60 percent to 35 percent. (Watch CBS News correspondent Nancy Cordes' report at left. ) In August, Mr. Obama's advantage among Ohio women was just 13 points. Ohio men, meanwhile, prefer Romney over Mr. Obama by a margin of eight points, 52 percent to 44 percent. In August, Romney held a two-point lead among Ohio men.

In Florida, women voters prefer Mr. Obama by 19 points, 58 percent to Romney's 39 percent. In August, the president's lead among Florida women was 12 points. Male voters in Florida prefer Romney, 50 percent to 47 percent. Romney's advantage among Florida men has diminished slightly since August, when he had a six-point lead.

It's impossible to predict what the gender gap in these states will ultimately look like on Election Day. In 2008, Mr. Obama won the support of both men and women in Ohio and Florida, though the gap was narrower among men. Nationally, Mr. Obama won the female vote 56 percent to 43 percent in 2008.

In 2004, Democrat John Kerry carried the female vote nationally, 51 percent to 48 percent, though he lost the election to George W. Bush.